Distributive laws between the operads Lie and Com
Murray Bremner, Vladimir Dotsenko
TL;DR
This paper classifies distributive laws between Lie and Com operads, revealing that the only nontrivial law is the Livernet-Loday formula, which deforms the Poisson operad into the associative operad.
Contribution
It provides a complete classification of distributive laws between Lie and Com operads using computer algebra methods, identifying the unique nontrivial law.
Findings
The Livernet-Loday formula is the only nontrivial distributive law between Lie and Com.
The classification is achieved through Gröbner bases in computer algebra.
The result links Poisson and associative operads via a unique deformation.
Abstract
Using methods of computer algebra, especially Gr\"obner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet-Loday formula deforming the Poisson operad into the associative operad.
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